I think you are confused by treating this as a standard quadratic which can only have two real roots or a pair of conjugate complex roots. But this is not. As others commented it is more appropriate to use calculus to detect the distribution of roots on the real line. In the complex case your equation become $$z^{2}-z\frac{e^{iz}-e^{-iz}}{2}-\frac{e^{iz}+e^{-iz}}{2}$$ this equation is trancendatal and probably do not have easy solutions.